William P. Thurston
The Geometry and Topology of Three-Manifolds
Electronic version 1.0 - October 1997
http://www.msri.org/gt3m/
This is an electronic edition of the 1980 notes distributed by Princeton University.
The text was typed in TEX by Sheila Newbery, who also scanned the figures. Typos
have been corrected (and probably others introduced), but otherwise no attempt has
been made to update the contents.
Numbers on the right margin correspond to the original edition’s page numbers.
Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University
Press, 1997) is a considerable expansion of the first few chapters of these notes. Later
chapters have not yet appeared in book form.
Please send corrections to Silvio Levy at [email protected].
CHAPTER 8
Kleinian groups
8.1
Our discussion so far has centered on hyperbolic manifolds which are closed, or
at least complete with finite volume. The theory of complete hyperbolic manifolds
with infinite volume takes on a somewhat different character. Such manifolds occur
very naturally as covering spaces of closed manifolds. They also arise in the study
of hyperbolic structures on compact three-manifolds whose boundary has negative
Euler characteristic. We will study such manifolds by passing back and forth between
the manifold and the action of its fundamental group on the disk.
8.1. The limit set
Let Γ be any discrete group of orientation-preserving isometries of H n. If x ∈ H n
n−1
is any point, the limit set LΓ ⊂ S∞
is defined to be the set of accumulation points
of the orbit Γx of x. One readily sees that LΓ is independent of the choice of x
by picturing the Poincaré disk model. If y ∈ H n is any other point and if {γi}
n−1
is a sequence of elements of Γ such that {γi x} converges to a point on S∞
, the
hyperbolic distance d(γi x, γi y) is constant so the Euclidean distance goes to 0; hence
lim γiy = lim γi x.
The group Γ is called elementary if the limit set consists of 0, 1 or 2 points.
Proposition 8.1.1. Γ is elementary if and only if Γ has an abelian subgroup of
finite index.
When Γ is not elementary, then LΓ is also the limit set of any orbit on the sphere
at infinity. Another way to put it is this:
Proposition 8.1.2. If Γ is not elementary, then every non-empty closed subset
of S∞ invariant by Γ contains LΓ .
Proof. Let K ⊂ S∞ be any closed set invariant by Γ. Since Γ is not elementary,
K contains more than one element. Consider the projective (Klein) model for H n ,
and let H(K) denote the convex hull of K. H(K) may be regarded either as the
Euclidean convex hull, or equivalently, as the hyperbolic convex hull in the sense
that it is the intersection of all hyperbolic half-spaces whose “intersection” with S∞
contains K. Clearly H(K) ∩ S∞ = K.
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8. KLEINIAN GROUPS
Since K is invariant by Γ, H(K) is also invariant by Γ. If x is any point in
H ∩ H(K), the limit set of the orbit Γx must be contained in the closed set H(K).
Therefore LΓ ⊂ K.
n
8.3
A closed set K invariant by a group Γ which contains no smaller closed invariant
set is called a minimal set. It is easy to show, by Zorn’s lemma, that a closed
invariant set always contains at least one minimal set. It is remarkable that in the
present situation, LΓ is the unique minimal set for Γ.
Corollary 8.1.3. If Γ is a non-elementary group and 1 6= Γ0 / Γ is a normal
subgroup, then LΓ0 = LΓ .
Proof. An element of Γ conjugates Γ0 to itself, hence it takes LΓ0 to LΓ0 . Γ0 must
be infinite, otherwise Γ0 would have a fixed point in H n which would be invariant by
Γ so Γ would be finite. It follows from 8.1.2 that LΓ0 ⊃ LΓ . The opposite inclusion
is immediate.
Examples. If M 2 is a hyperbolic surface, we may regular π1 (M) as a group of
isometries of a hyperbolic plane in H 3 . The limit set is a circle. A group with limit
set contained in a geometric circle is called a Fuchsian group.
n−1
The limit set for a closed hyperbolic manifold is the entire sphere S∞
.
If M 3 is a closed hyperbolic three-manifold which fibers over the circle, then
the fundamental group of the fiber is a normal subgroup, hence its limit set is the
entire sphere. For instance, the figure eight knot complement has fundamental group
hA, B : ABA−1 BA = BAB −1 ABi.
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8.4
8.1. THE LIMIT SET
It fibers over S 1 with fiber F a punctured torus. The fundamental group π1 (F )
is the commutator subgroup, generated by AB −1 and A−1 B. Thus, the limit set of
a finitely generated group may be all of S 2 even when the quotient space does not
have finite volume.
A more typical example of a free group action is a Schottky group, whose limit
set is a Cantor set. Examples of Schottky groups may be obtained by considering
H n minus 2k disjoint half-spaces, bounded by hyperplanes. If we choose isometric
identifications between pairs of the bounding hyperplanes, we obtain a complete
hyperbolic manifold with fundamental group the free group on k generators.
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8. KLEINIAN GROUPS
8.5
It is easy to see that the limit set for the group of covering transformations is a
Cantor set.
8.2. The domain of discontinuity
n−1
The domain of discontinuity for a discrete group Γ is defined to be DΓ = S∞
−LΓ .
A discrete subgroup of PSL(2, C ) whose domain of discontinuity is non-empty is called
a Kleinian group. (There are actually two ways in which the term Kleinian group is
generally used. Some people refer to any discrete subgroup of PSL(2, C ) as a Kleinian
2
, and a type II
group, and then distinguish between a type I group, for which LΓ = S∞
group, where DΓ 6= ∅. As a field of mathematics, it makes sense for Kleinian groups
to cover both cases, but as mathematical objects it seems useful to have a word to
distinguish between these cases DΓ 6= ∅ and DΓ = ∅.)
We have seen that the action of Γ on LΓ is minimal—it mixes up LΓ as much as
possible. In contrast, the action of Γ on DΓ is as discrete as possible.
Definition 8.2.1. If Γ is a group acting on a locally compact space X, the action
is properly discontinuous if for every compact set K ⊂ X, there are only finitely many
γ ∈ Γ such that γK ∩ K 6= ∅.
Another way to put this is to say that for any compact set K, the map Γ×K → X
given by the action is a proper map, where Γ has the discrete topology. (Otherwise
there would be a compact set K 0 such that the preimage of K 0 is non-compact. Then
infinitely many elements of Γ would carry K ∪ K 0 to itself.)
Proposition 8.2.2. If Γ acts properly discontinuously on the locally compact
Hausdorff space X, then the quotient space X is Hausdorff. If the action is free,
the quotient map X → X/Γ is a covering projection.
Proof. Let x1 , x2 ∈ X be points on distinct orbits of Γ. Let N1 be a compact
neighborhood of x1 . Finitely many translates
S of x2 intersect N1 , so we may assume
N1 is disjoint from the orbit of x2 . Then γ∈Γ γN1 gives an invariant neighborhood
of x1 disjoint from x2 . Similarly, x2 has an invariant neighborhood N2 disjoint from
N1 ; this shows that X/Γ is Hausdorff. If the action of Γ is free, we may find,
again by a similar argument, a neighborhood of any point x which is disjoint from
all its translates. This neighborhood projects homeomorphically to X/Γ. Since Γ
acts transitively on the sheets of X over X/Γ, it is immediate that the projection
X → X/Γ is an even covering, hence a covering space.
Proposition 8.2.3. If Γ is a discrete group of isometries of H n , the action of Γ
on DΓ (and in fact on H n ∪ DΓ ) is properly discontinuous.
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8.6
8.2. THE DOMAIN OF DISCONTINUITY
Proof. Consider the convex hull H(LΓ ). There is a retraction r of the ball
H ∪ S∞ to H(LΓ ) defined as follows.
If x ∈ H(LΓ ), r(x) = x. Otherwise, map x to the nearest point of H(LΓ ). If
x is an infinite point in DΓ , the nearest point is interpreted to be the first point of
H(LΓ ) where a horosphere “centered” about x touches LΓ . This point r(x) is always
uniquely defined
n
because H(LΓ ) is convex, and spheres or horospheres about a point in the ball are
strictly convex. Clearly r is a proper map of H n ∪ DΓ to H(LΓ ) − LΓ . The action of
Γ on H(LΓ ) − LΓ is obviously properly discontinuous, since Γ is a discrete group of
isometries of H(LΓ ) − LΓ ; the property of H n ∪ DΓ follows immediately.
Remark. This proof doesn’t work for certain elementary groups; we will ignore
such technicalities.
It is both easy and common to confuse the definition of properly discontinuous
with other similar properties. To give two examples, one might make these definitions:
Definition 8.2.4. The action of Γ is wandering if every point has a neighborhood N such that only finitely many translates of N intersect N.
Definition 8.2.5. The action of Γ has discrete orbits if every orbit of Γ has an
empty limit set.
Proposition 8.2.6. If Γ is a free, wandering action on a Hausdorff space X, the
projection X → X/Γ is a covering projection.
Proof. An exercise.
Warning. Even when X is a manifold, X/Γ may not be Hausdorff. For instance,
consider the map
L : R2 − 0 → R2 − 0
L (x, y) = (2x, 12 y).
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8.7
8. KLEINIAN GROUPS
It is easy to see this is a wandering action. The quotient space is a surface with
fundamental group Z ⊕ Z. The surface is non-Hausdorff, however, since points such
as (1, 0) and (0, 1) do not have disjoint neighborhoods.
Such examples arise commonly and naturally; it is wise to be aware of this phenomenon.
The property that Γ has discrete orbits simply means that for every pair of points
x, y in the quotient space X/Γ, x has a neighborhood disjoint from y. This can occur,
for instance, in a l-parameter family of Kleinian groups Γt , t ∈ [0, 1]. There are
examples where Γt = Z, and the family defines the action of Z on [0, 1] × H 3 with
discrete orbits which is not a wandering action. See § . It is remarkable that the
action of a Kleinian group on the set of all points with discrete orbits is properly
discontinuous.
8.3. Convex hyperbolic manifolds
The limit set of a group action is determined by a limiting process, so that it
is often hard to “know” the limit set directly. The condition that a given group
action is discrete involves infinitely many group elements, so it is difficult to verify
directly. Thus it is important to have a concrete object, satisfying concrete conditions,
corresponding to a discrete group action.
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8.9
8.3. CONVEX HYPERBOLIC MANIFOLDS
We consider for the present only groups acting freely.
Definition 8.3.1. A complete hyperbolic manifold M with boundary is convex
if every path in M is homotopic (rel endpoints) to a geodesic arc. (The degenerate
case of an arc which is a single point may occur.)
8.10
Proposition 8.3.2. A complete hyperbolic manifold M is convex if and only if
the developing map D : M̃ → H n is a homeomorphism to a convex subset of H n .
Proof. If M̃ is a convex subset S of H n, then it is clear that M is convex, since
any path in M lifts to a path in S, which is homotopic to a geodesic arc in S, hence
in M.
If M is convex, then D is 1 − 1, since any two points in M̃ may be joined by a
path, which is homotopic in M and hence in M̃ to a geodesic arc. D must take the
endpoints of a geodesic arc to distinct points. D(M̃ ) is clearly convex.
We need also a local criterion for M to be convex. We can define M to be locally
convex if each point
x ∈ M has a neighborhood isometric to a convex subset of H n . If x ∈ ∂M, then x
will be on the boundary of this set. It is easy to convince oneself that local convexity
implies convexity: picture a bath and imagine straightening it out. Because of local
convexity, one never needs to push it out of ∂M. To make this a rigorous argument,
given a path p of length l there is an such that any path of length ≤ intersecting
Nl (p0) is homotopic to a geodesic arc. Subdivide p into subintervals of length between
/4 and /2. Straighten out adjacent pairs of intervals in turn, putting a new division
point in the middle of the resulting arc unless it has length ≤ /2. Any time an
interval becomes too small, change the subdivision. This process converges, giving a
homotopy of p to a geodesic arc, since any time there are angles not close to π, the
homotopy significantly shortens the path.
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8.11
8. KLEINIAN GROUPS
This give us a very concrete object corresponding to a Kleinian group: a complete
convex hyperbolic three-manifold M with non-empty boundary.
Given a convex manifold M, we can define H(M) to be the intersection of all
convex submanifolds M 0 of M such that π1 M 0 → π1 M is an isomorphism. H(M) is
clearly the same as HLπ1 (M)/π1 (M). H(M) is a convex manifold, with the same
dimension as M except in degenerate cases.
Proposition 8.3.3. If M is a compact convex hyperbolic manifold, then any
small deformation of the hyperbolic structure on M can be enlarged slightly to give a
new convex hyperbolic manifold homeomorphic to M.
Proof. A convex manifold is strictly convex if every geodesic arc in M has interior in the interior of M. If M is not already strictly convex, it can be enlarged
slightly to make it strictly convex. (This follows from the fact that a neighborhood
of radius about a hyperplane is strictly convex.)
Thus we may assume that M 0 is a hyperbolic structure that is a slight deformation
of a strictly convex manifold M. We may assume that our deformation M 0 is small
enough that it can be enlarged to a hyperbolic manifold M 00 which contains a 2neighborhood of M 0 . Every arc of length l greater than in M has the middle (l − )
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8.12
8.3. CONVEX HYPERBOLIC MANIFOLDS
some uniform distance δ from ∂M; we may take our deformation M 0 of M small
enough that such intervals in M 0 have the middle l − still in the interior of M 0 . This
implies that the union of the convex hulls of intersections of balls of radius 3 with
M 0 is locally convex, hence convex.
8.13
The convex hull of a uniformly small deformation of a uniformly convex manifold
is locally determined.
Remark. When M is non-compact, the proof of 8.3.3 applies provided that M
has a uniformly convex neighborhood and we consider only uniformly small deformations. We will study deformations in more generality in § .
Proposition 8.3.4. Suppose M1n and M2n are strictly convex, compact hyperbolic
manifolds and suppose φ : M1n → M2n is a homotopy equivalence which is a diffeomorphism on ∂M1 . Then there is a quasi-conformal homeomorphism f : B n → B n
of the Poincaré disk to itself conjugating π1 M1 to π1 M2 . f is a pseudo-isometry on
H n.
Proof. Let φ̃ be a lift of φ to a map from M̃1 to M̃2 . We may assume that φ̃ is
already a pseudo-isometry between the developing images of M1 and M2 . Each point
p on ∂ M̃1 and ∂ M̃2 has a unique normal ray γp ; if x ∈ γp has distance t from ∂ M̃1
let f (x) be the point on γφ̃(p) a distance t from ∂ M̃2 . The distance between points at
a distance of t along two normal rays γp1 and γp2 at nearby points is approximately
cosh t + α sinh t, where d is the distance and θ is the angle between the normals of p1
and p2 . From this it is evident that f is a pseudo-isometry extending to φ̄.
Associated with a discrete group Γ of isometries of H n , there are at least four
distinct and interesting quotient spaces (which are manifolds when Γ acts freely ).
Let us name them:
Definition 8.3.5.
MΓ = H(LΓ )/Γ , the convex hull quotient.
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8.14
8. KLEINIAN GROUPS
NΓ = H n/Γ, the complete hyperbolic manifold without boundary.
OΓ = (H n ∪ DΓ )/Γ, the Kleinian manifold.
PΓ = (H n ∪ DΓ ∪ WΓ )/Γ. Here WΓ ⊂ Pn is the set of points in the projective
model dual to planes in H n whose intersection with S∞ is contained in DΓ .
We have inclusions H(NΓ ) = MΓ ⊂ NΓ ⊂ OΓ ⊂ PΓ . It is easy to derive the fact that
Γ acts properly discontinuously on H n ∪ DΓ ∪ WΓ from the proper discontinuity on
H n ∪ DΓ . MΓ , NΓ and OΓ have the same homotopy type. MΓ and OΓ are homeomorphic except in degenerate cases, and NΓ = int(OΓ ) PΓ is not always connected
when LΓ is not connected.
8.15
8.4. Geometrically finite groups
Definition 8.4.1. Γ is geometrically finite if
N(MΓ) has finite volume.
The reason that N (MΓ ) is required to have finite volume, and not just MΓ , is to
rule out the case that Γ is an arbitary discrete group of isometries of H n−1 ⊂ H n .
We shall soon prove that geometrically finite means geometrically finite (8.4.3).
Theorem 8.4.2 (Ahlfors’ Theorem). If Γ is geometrically finite, then LΓ ⊂ S∞
has full measure or 0 measure. If LΓ has full measure, the action of Γ on S∞ is
ergodic.
Proof. This statement is equivalent to the assertion that every bounded measurable function f supported on LΓ and invariant by Γ is constant a.e. (with respect
to Lebesque measure on S∞ ). Following Ahlfors, we consider the function hf on H n
determined by f as follows. If x ∈ H n , the points on S∞ correspond to rays through
x; these rays have a natural “visual” measure Vx . Define hf (x) to be the average
of f with respect to the visual measure Vx . This function hf is harmonic, i.e., the
gradient flow of hf preserves volume,
div grad hf = 0.
For this reason, the measure Vx (S1 ∞ ) Vx is called harmonic measure. To prove this,
consider the contribution to hf coming from an infinitesimal area A centered at
p ∈ S n−1 (i.e., a Green’s function). As x moves a distance d in the direction of
p, the visual measure of A goes up exponentially, in proportion to e(n−1)d . The
gradient of any multiple of the characteristic function of A is in the direction of p,
and also proportional in size to e(n−1)d . The flow lines of the gradient are orthogonal
trajectories to horospheres; this flow contracts linear dimensions along the horosphere
in proportion to e−d , so it preserves volume.
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8.16
8.4. GEOMETRICALLY FINITE GROUPS
The average hf of contributions from all the infinitesimal areas is therefore harmonic.
We may suppose that f takes only the values of 0 and 1. Since f is invariant by Γ,
so is hf , and hf goes over to a harmonic function, also hf , on NΓ . To complete the
proof, observe that hf < 12 in NΓ − MΓ , since each point x in H n − H(LΓ ) lies in
a half-space whose intersection with infinity does not meet LΓ , which means that f
is 0 on more than half the sphere, with respect to Vx . The set {x ∈ NΓ |hf (x) = 12 }
must be empty, since it bounds the set {x ∈ NΓ |hf (x) ≥ 12 } of finite volume which
flows into itself by the volume preserving flow generated by grad hf . (Observe that
grad hf has bounded length, so it generates a flow defined everywhere for all time.)
But if {p|f (p) = 1} has any points of density, then there are x ∈ H n−1 near p with
hf (x) near 1. It follows that f is a.e. 0 or a.e. 1.
Let us now relate definition 8.4.1 to other possible notions of geometric finiteness.
The usual definition is in terms of a fundamental polyhedron for the action of Γ.
For concreteness, let us consider only the case n = 3. For the present discussion, a
finite-sided polyhedron means a region P in H 3 bounded by finitely many planes. P
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8.17
8. KLEINIAN GROUPS
is a fundamental polyhedron for Γ if its translates by Γ cover H 3 , and the translates
of its interior are pairwise disjoint. P intersects S∞ in a polygon which unfortunately
may be somewhat bizarre, since tangencies between sides of P ∩ S∞ may occur.
Sometimes these tangencies are forced by the existence of parabolic fixed points
for Γ. Suppose that p ∈ S∞ is a parabolic fixed point for some element of Γ, and let π
be the subgroup of Γ fixing p. Let B be a horoball centered at p and sufficiently small
that the projection of B/P to NΓ is an embedding. (Compare §5.10.) If π ⊃ Z ⊕ Z,
for any point x ∈ B ∩ H(LΓ ), the convex hull of πx contains a horoball B 0 , so in
particular there is a horoball B 0 ⊂ H(LΓ ) ∩ B. Otherwise, Z is a maximal torsionfree subgroup of π. Coordinates can be chosen so that p is the point at ∞ in the
upper half-space model, and Z acts as translations by real integers. There is some
minimal strip S ⊆ C containing LΓ ∩ C ; S may interesect the imaginary axis in a
finite, half-infinite, or doubly infinite interval. In any case, H(LΓ ) is contained in the
region R of upper half-space above S, and the part of ∂R of height ≥ 1 lies on ∂HΓ .
It may happen that there are wide substrips S 0 ⊂ S in the complement of LΓ . If S 0
is sufficiently wide, then the plane above its center line intersects H(LΓ ) in B, so it
gives a half-open annulus in B/Z. If Γ is torsion-free, then maximal, sufficiently wide
strips in S − LΓ give disjoint non-parallel half-open annuli in MΓ ; an easy argument
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8.18
8.19
8.4. GEOMETRICALLY FINITE GROUPS
shows they must be finite in number if Γ is finitely generated. (This also follows from
Ahlfors’s finiteness theorem.) Therefore, there is some horoball B 0 centered at p so
that H(LΓ ) ∩ B 0 = R ∩ B 0 . This holds even if Γ has torsion.
8.20
With an understanding of this picture of the behaviour of MΓ near a cusp, it is
not hard to relate various notions of geometric finiteness. For convenience suppose Γ
is torsion-free. (This is not an essential restriction in view of Selberg’s theorem—see
§ .) When the context is clear, we abbreviate MΓ = M, NΓ = N, etc.
Proposition 8.4.3. Let Γ ⊂ PSL(2, C ) be a discrete, torsion-free group. The
following conditions are equivalent:
(a) Γ is geometrically finite (see dfn. 8.4.1).
(b) M[,∞) is compact.
(c) Γ admits a finite-sided fundamental polyhedron.
Proof. (a) ⇒ (b).
Each point in M[,∞) has an embedded /2 ball in N/2(MΓ ), by definition. If (a)
holds, N/n (MΓ ) has finite volume, so only finitely many of these balls can be disjoint
and MΓ[,∞) is compact.
(b) ⇒ (c). First, find fundamental polyhedra near the non-equivalent parabolic
fixed points. To do this, observe that if p is a Z-cusp, then in the upper half-space
model, when p = ∞, LΓ ∩ C lies in a strip S of finite width. Let R denote the region
above S. Let B 0 be a horoball centered at ∞ such that R ∩ B 0 = H(LΓ ) ∩ B 0 . Let
r : H 3 ∪ DΓ → H(LΓ ) be the canonical retraction. If Q is any fundamental polyhedra
for the action of Z in some neighborhood of p in H(LΓ ) then r−1 (Q) is a fundamental
polyhedron in some neighborhood of p in H 3 ∪ DΓ .
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8.21
8. KLEINIAN GROUPS
A fundamental polyhedron near the cusps is easily extended to a global fundamental
polyhedron, since OΓ -(neighborhoods of the cusps) is compact.
(c) ⇒ (a). Suppose that Γ has a finite-sided fundamental polyhedron P .
A point x ∈ P ∩ S∞ is a regular point (∈ DΓ ) if it is in the interior of P ∩ S∞
or of some finite union of translates of P . Thus, the only way x can be a limit point
is for x to be a point of tangency of sides of infinitely many translates of P . Since
P can have only finitely many points of tangency of sides, infinitely many γΓ must
identify one of these points to x, so x is a fixed point for some element γΓ. γ must
be parabolic, otherwise the translates of P by powers of γ would limit on the axis
of γ. If x is arranged to be ∞ in upper half-space, it is easy to see that LΓ C must
be contained in a strip of finite width. (Finitely many translates of P must form a
fundamental domain for {γ n}, acting on some horoball centered at ∞, since {γ n } has
finite index in the group fixing ∞. Th faces of these translates of P which do not
pass through ∞ lie on hemispheres. Every point in C outside this finite collection of
hemispheres and their translates by {γ n} lies in DΓ .)
It follows that v(N (M)) = v(N(H(LΓ )) ∩ P ) if finite, since the contribution near
any point of LΓ ∩ P is finite and the rest of N (H(LΓ )) ∩ P is compact.
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8.22
8.5. THE GEOMETRY OF THE BOUNDARY OF THE CONVEX HULL
8.5. The geometry of the boundary of the convex hull
Consider a closed curve σ in Euclidean space, and its convex hull H(σ). The
boundary of a convex body always has non-negative Gaussian curvature. On the
other hand, each point p in ∂H(σ) − σ lies in the interior of some line segment or
triangle with vertices on σ. Thus, there is some line segment on ∂H(σ) through p,
so that ∂H(σ) has non-positive curvature at p. It follows that ∂H(σ) − σ has zero
curvature, i.e., it is “developable”. If you are not familiar with this idea, you can
see it by bending a curve out of a piece of stiff wire (like a coathanger). Now roll
the wire around on a big piece of paper, tracing out a curve where the wire touches.
Sometimes, the wire may touch at three or more points; this gives alternate ways
to roll, and you should carefully follow all of them. Cut out the region in the plane
bounded by this curve (piecing if necessary). By taping the paper together, you can
envelope the wire in a nice paper model of its convex hull. The physical process
of unrolling a developable surface onto the plane is the origin of the notion of the
developing map.
The same physical notion applies in hyperbolic three-space. If K is any closed
set on S∞ , then H(K) is convex, yet each point on ∂H(K) lies on a line segment
in ∂H(K). Thus, ∂H(K) can be developed to a hyperbolic plane. (In terms of
Riemannian geometry, ∂H(K) has extrinsic curvature 0, so its intrinsic curvature
is the ambient sectional curvature, −1. Note however that ∂H(K) is not usually
differentiable). Thus ∂H(K) has the natural structure of a complete hyperbolic
surface.
8.23
Proposition 8.5.1. If Γ is a torsion-free Kleinian group, the ∂MΓ is a hyperbolic
surface.
The boundary of MΓ is of course not generally flat—it is bent in some pattern.
Let γ ⊂ ∂MΓ consist of those points which are not in the interior of a flat region of
∂MΓ . Through each point x in γ, there is a unique geodesic gx on ∂MΓ . gx is also a
geodesic in the hyperbolic structure of ∂MΓ . γ is a closed set. If ∂MΓ has finite area,
then γ is compact, since a neighborhood of each cusp of ∂MΓ is flat. (See §8.4.)
Definition 8.5.2. A lamination L on a manifold M n is a closed subset A ⊂ M
(the support of L) with a local product structure for A. More precisely, there is a
φi
covering of a neighborhood of A in M with coordinate neighborhoods Ui → R n−k × R k
so that φi (A ∩ Ui ) is of the form R n−k × B, B ⊂ R k . The coordinate changes φij
must be of the form φij (x, y) = (fij (x, y), gij (y)) when y ∈ B. A lamination is like a
foliation of a closed subset of M. Leaves of the lamination are defined just as for a
foliation.
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8. KLEINIAN GROUPS
Examples. If F is a foliation of M and S ⊂ M is any set, the closure of the
union of leaves which meet S is a lamination.
Any submanifold of a manifold M is a lamination, with a single leaf.
Clearly, the bending locus γ for ∂MΓ has the structure of a lamination: whenever
two points of γ are nearby, the directions of bending must be nearly parallel in order
that the lines of bending do not intersect. A lamination whose leaves are geodesics
we will call a geodesic lamination.
8.25
By consideration of Euler characteristic, the lamination γ cannot have all of ∂M
as its support, or in other words it cannot be a foliation. The complement ∂M − γ
consists of regions bounded by closed geodesics and infinite geodesics. Each of these
regions can be doubled along its boundary to give a complete hyperbolic surface,
which of course has finite area. There
8.26
is a lower bound for π for the area of such a region, hence an upper bound of
2|χ(∂M)| for the number of components of ∂M − γ. Every geodesic lamination γ on
a hyperbolic surface S can be extended to a foliation with isolated singularities on
the complement. There
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8.5. THE GEOMETRY OF THE BOUNDARY OF THE CONVEX HULL
is an index formula for the Euler characteristic of S in terms of these singularities.
Here are some values for the index.
8.27
From the existence of an index formula, one concludes that the Euler characteristic
of S is half the Euler characteristic of the double of S − γ. By the Gauss-Bonnet
theorem,
Area(S − γ) = Area(S)
or in other words, γ has measure 0.
To give an idea of the range of possibilities for geodesic laminations, one can
consider an arbitrary sequence {γi } of geodesic laminations: simple closed curves, for
instance. Let us say that {γi } converges geometrically to γ if for each x ∈ support γ,
and for each , for all great enough i the support of γi intersects N(x) and the leaves
of γi ∩ N (x) are within of the direction of the leaf of γ through x. Note that the
support of γ may be smaller than the limiting support of γi , so the limit of a sequence
may not be unique. See §8.10. An easy diagonal argument shows that every sequence
{γi } has a subsequence which converges geometrically. From limits of sequences of
simple closed geodesics, uncountably many geodesic laminations are obtained.
Geodesic laminations on two homeomorphic hyperbolic surfaces may be compared
by passing to the circle at ∞. A directed geodesic is determined by a pair of points
1
1
(x1 , x2 ) ∈ S∞
× S∞
− ∆, where ∆ is the diagonal {(x, x)}. A geodesic without
1
1
direction is a point on J = (S∞
× S∞
− ∆/Z2), where Z2 acts by interchanging
coordinates. Topologically, J is an open Moebius band. It is geometrically realized
in the Klein (projective) model for H 2 as the region outside H 2 . A geodesic g projects
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8.28
8. KLEINIAN GROUPS
to a simple geodesic on the surface S if and only if the covering translates of its pairs
of end points never strictly separate each other.
Geometrically, J has an indefinite metric of type (1, 1), invariant by covering
translates. (See §2.6.) The light-like geodesics, of zero length, are lines tangent to
1
S∞
; lines which meet H 2 when extended have imaginary arc length. A point g ∈ J
projects to a simple geodesic in S if and only if no covering translate Tα (g) has a
positive real distance from g.
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8.6. MEASURING LAMINATIONS
Let S ⊂ J consist of all elements g projecting to simple geodesics on S. Any
geodesic ⊂ H 2 which has a translate intersecting itself has a neighborhood with the
same property, hence S is closed.
If γ is any geodesic lamination on S, Let Sγ ⊂ J be the set of lifts of leaves of
γ to H 2 . Sγ is a closed invariant subset of S. A closed invariant subset of C ⊂ J
gives rise to a geodesic lamination if and only if all pairs of points of C are separated
by an imaginary (or 0) distance. If g ∈ S, then the closure of its orbit, π1 (S)g is
such a set, corresponding to the geodesic lamination ḡ of S. Every homeomorphism
1
between surfaces when lifted to H 2 extends to S∞
(by 5.9.5). This determines an
extension to J. Geodesic laminations are transferred from one surface to another via
this correspondence.
8.29
8.6. Measuring laminations
Let L be a lamination, so that it has local homeomorphisms φi : L∩Ui ≈ R n−k ×Bi .
A transverse measure µ for L means a measure µi defined on each local leaf space Bi ,
in such a way that the coordinate changes are measure preserving. Alternatively one
may think of µ as a measure defined on every k-dimensional submanifold transverse
to L, supported on T k ∩ L and invariant under local projections along leaves of L.
We will always suppose that µ is finite on compact transversals.
The simplest example of a transverse measure arises when L is a closed submanifold; in this case, one can take µ to count the number of intersections of a transversal
with L.
We know that for a torsion-free Kleinian group Γ, ∂MΓ is a hyperbolic surface
bent along some geodesic lamination γ. In order to complete the picture of ∂MΓ ,
we need a quantitative description of the bending. When two planes in H 3 meet
along a line, the angle they form is constant along that line. The flat pieces of ∂MΓ
meet each other along the geodesic lamination γ; the angle of meeting of two planes
generalizes to a transverse “bending” measure, β, for γ. The measure β applied
to an arc α on ∂MΓ transverse to γ is the total angle of turning of the normal to
∂MΓ along α (appropriately interpreted when γ has isolated geodesics with sharp
bending). In order to prove that β is well-defined, and that it determines the local
isometric embedding in H 3 , one can use local polyhedral approximations to ∂MΓ .
Local outer approximations to ∂MΓ can be obtained by extending the planes of local
flat regions. Observe that when three planes have pairwise intersections in H 3 but
no triple intersection, the dihedral angles satisfy the inequality
α + β ≤ γ.
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8.30
8. KLEINIAN GROUPS
8.31
(The difference γ − (α + β) is the area of a triangle on the common perpendicular
plane.) From this it follows that as outer polyhedral approximations shrink toward
MΓ , the angle sum corresponding to some path α on ∂MΓ is a monotone sequence,
converging to a value β(α). Also from the monotonicity, it is easy to see that for
short paths αt , [0 ≤ t ≤ 1], β(α) is a close approximation to the angle between the
tangent planes at α0 and α1 . This implies that the hyperbolic structure on ∂MΓ ,
together with the geodesic lamination γ and the transverse measure β, completely
determines the hyperbolic structure of NΓ in a neighborhood of ∂MΓ .
The bending measure β has for its support all of γ. This puts a restriction on the
structure of γ: every isolated leaf L of γ must be a closed geodesic on ∂MΓ . (Otherwise, a transverse arc through any limit point of L would have infinite measure.) This
limits the possibilities for the intersection of a transverse arc with γ to a Cantor set
and/or a finite set of points.
When γ contains more than one closed geodesic, there is obviously a whole family
of possibilities for transverse measures. There are (probably atypical) examples of
families of distinct transverse measures which are not multiples of each other even for
certain geodesic laminations such that every leaf is dense. There are many other examples which possess unique transverse measures, up to constant multiples. Compare
Katok.
Here is a geometric interpretation for the bending measure β in the Klein model.
Let P0 be the component of PΓ containing NΓ (recall definition 8.3.5). Each point in
P̃0 outside S∞ is dual to a plane which bounds a half-space whose intersection with
S∞ is contained in DΓ . ∂ P̃0 consists of points dual to planes which meet LΓ in at
least one point. In particular, each plane meeting M̃Γ in a line or flat of ∂ M̃Γ is dual
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8.32